Statistics on Correlation and Regression Models

Learning Objectives

• Compute and interpret Pearson correlation coefficient.
• Compute and interpret Spearman rank correlation coefficient.
• Describe the purpose and use of linear regression models.
• Calculate a simple linear regression model for two related variables.

Scatter Diagrams

• Visual tool for displaying relationships between two variables (X and Y).
• Illustrates direction and strength of associations.
• Less effective for large sample sizes due to point densely crowding.

Correlation

• Numerical measure of the linear relationship between two quantitative variables.
• Indicates how closely two variables are associated.
• Common examples include height and weight, age and heart disease, and body mass index and blood pressure.

Correlation Coefficients

• Calculated using:
  • Pearson product moment correlation coefficient.
  • Spearman's rank correlation coefficient.

Pearson Correlation Coefficient

• Developed by Karl Pearson, measures linear association strength between continuous variables.
• Values range from -1 to +1:
  • +1: perfect positive correlation.
  • -1: perfect negative correlation.
  • 0: no linear correlation exists.

Interpretation of Pearson's Correlation Coefficient

• Positive values indicate that as one variable increases, the other tends to increase.
• Negative values indicate that as one variable increases, the other tends to decrease.
• The coefficient's magnitude reflects the strength of the relationship, with values close to -1 or +1 indicating stronger correlations.

Coefficient of Determination

• The square of the correlation coefficient (r²) indicates the proportion of variability in the response variable due to the predictor variables.
• Example: An r² value of 0.3871 suggests that about 38.71% of variability in the response can be explained.

Linear Regression

• A statistical method used to summarize relationships between numeric variables.
• Includes types like simple linear regression, multiple linear regression, and logistic regression.

Simple Linear Regression

• Quantifies association between two variables with a single predictor.
• Aims to estimate the relationship form between variables.
• Best fit line is represented mathematically by the equation: Y = β0 + β1(X), where:
  • β0: intercept
  • β1: slope of the line

Practical Example in Regression

• To determine the relationship between systolic blood pressure (SBP) and diastolic blood pressure (DBP):
  • Collect data from patients.
  • Use linear regression to analyze the relationship and predict DBP from SBP.

Summary Points

• The sign and magnitude of correlation coefficients provide insight into variable relationships.
• Linear regression serves as a foundational analytical approach for exploring dependencies between variables.
• Understanding coefficients and their interpretations is crucial for data analysis in epidemiology and medical statistics.

This quiz assesses your understanding of correlation coefficients such as Pearson and Spearman, and their application in linear regression models. You will learn how to compute, interpret, and visualize relationships between two quantitative variables through scatter diagrams and correlation analysis.